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Scattering of Nucleons in the Classical Skyrme Model Foster, D., & Manton, N. S. (2015). Scattering of Nucleons in the Classical Skyrme Model. Nuclear Physics B, 899, 513-526. https://doi.org/10.1016/j.nuclphysb.2015.08.012 Publisher's PDF, also known as Version of record License (if available): CC BY Link to published version (if available): 10.1016/j.nuclphysb.2015.08.012 Link to publication record in Explore Bristol Research PDF-document This is the final published version of the article (version of record). It first appeared online via Elsevier at 10.1016/j.nuclphysb.2015.08.012. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 899 (2015) 513–526 www.elsevier.com/locate/nuclphysb Scattering of nucleons in the classical Skyrme model ∗ ∗ David Foster a, , Nicholas S. Manton b, a School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK b Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Received 2 July 2015; accepted 13 August 2015 Available online 18 August 2015 Editor: Stephan Stieberger Abstract Classically spinning B = 1 Skyrmions can be regarded as approximations to nucleons with quantised spin. Here, we investigate nucleon–nucleon scattering through numerical collisions of spinning Skyrmions. We identify the dineutron/diproton and dibaryon short-lived resonance states, and also the stable deuteron state. Our simulations lead to predictions for the polarisation states occurring in right angle scattering. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The Skyrme model [1] is a nonlinear theory of pion fields that has topological soliton solu- tions. Encouragingly, Witten identified the Skyrme model as a low energy effective model of QCD [2,3]. The conserved topological charge is interpreted as the baryon number B, and the minimal energy static solutions for each integer B are called Skyrmions. They can be treated as rigid bodies, free to rotate in space and also in isospace (the three-dimensional space of the pion fields). When the rotational motion is quantised, the Skyrmions are models for nucleons and nuclei. In particular the B = 1 Skyrmions, quantised with spin and isospin half, model protons and neutrons. * Corresponding authors. E-mail addresses: [email protected] (D. Foster), [email protected] (N.S. Manton). http://dx.doi.org/10.1016/j.nuclphysb.2015.08.012 0550-3213/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 514 D. Foster, N.S. Manton / Nuclear Physics B 899 (2015) 513–526 A full quantum mechanical treatment of the interaction of two B = 1 Skyrmions is dif- ficult. Each Skyrmion has three position coordinates and three orientational coordinates, so two-Skyrmion dynamics involves twelve coordinates [4]. A truncation to ten coordinates has been useful for modelling the deuteron bound state [5], but no proper discussion of two-nucleon scattering is possible with this truncation. An alternative is a classical approach, using the idea that a classically spinning Skyrmion is a reasonable model of a nucleon [6,7]. The classical angular velocity of the Skyrmion is fixed to match the quantised spin and isospin of the nucleon. Scattering of non-spinning Skyrmions was first performed using an axially symmetric ansatz [8], the first full-field simulation was reported in [9], and multi-charge Skyrmion scattering was considered in [10]. The scattering of spinning Skyrmions representing nucleons was considered by Gisiger and Paranjape [6], and they obtained analytical formulae for the scattering angle for large impact parameters. There has also been some recent work on multi-charge, purely isospinning Skyrmions [12]. A systematic, numerical investigation of Skyrmion scattering without spin, at moderate and small impact parameters [11], confirmed that matter is exchanged between the Skyrmions when the Skyrmions are close, and also showed that a substantial rotation of the Skyrmion’s orientation can occur. Here, we numerically investigate the scattering of two classically spinning B = 1 Skyrmions to model proton–proton, neutron–neutron and proton–neutron scattering at various impact parameters, including head-on collisions. We are particularly interested in the change in the polarisation states of the nucleons when they scatter, and also in finding evidence, using our classical approximation, for the deuteron and for the known two-nucleon resonance states. The format of this paper is as follows. We first introduce the Skyrme model and its calibration. Then we discuss how to identify classically spinning Skyrmions as protons or neutrons. The next section presents the results of our numerical scattering of Skyrmions, and our identification of the dineutron/diproton resonances, the deuteron bound state, and the excited dibaryon state. The concluding section summarises and discusses our results. 2. The Skyrme model The Skyrme model is defined by the Lagrangian density 2 2 2 Fπ μ 1 μ ν mπ Fπ L =− Tr(RμR ) + Tr([Rμ,Rν][R ,R ]) + Tr(U − I ), (1) 16 32e2 8 2 † where the Skyrme field U(t, x) is an SU(2)-valued scalar and Rμ = ∂μUU is its su(2)-valued current. Fπ , e and mπ are parameters, which are fixed by comparison with experimental data. Their values will be discussed later. It is convenient for us to work in dimensionless Skyrme units. One Skyrme length unit corresponds to 2 in inverse MeV, and one Skyrme energy unit corre- eFπ Fπ ¯ = sponds to 4e MeV. Conversion of inverse MeV to fm is as usual through h 197.3MeVfm. The dimensionless pion mass in Skyrme units is 2 m = mπ . (2) eFπ In Skyrme units, the energy of a static field is 1 1 E = − Tr(R R ) − Tr([R ,R ][R ,R ]) + m2 Tr(I − U) d3x. 2 i i 16 i j i j 2 D. Foster, N.S. Manton / Nuclear Physics B 899 (2015) 513–526 515 Fig. 1. B = 1 Hedgehog Skyrmion. (For a colour image, we refer the reader to the web version of the article.) It is often convenient, especially in numerical simulations, to express U in terms of a triplet of pion fields π = (π1, π2, π3) and an additional auxiliary field σ as U(t,x) = σ(t,x)I2 + iπ(t, x) · τ , (3) 2 where τ = (τ1, τ2, τ3) are the three Pauli matrices, with the constraint σ + π · π = 1so that U ∈ SU(2). 3 At fixed time, U(t, x) is a map U : R → SU(2), where U → I2 at spatial infinity. This boundary condition compactifies R3 ∪{∞}to S3. The group manifold of SU(2) is S3, so a finite 3 3 3 energy configuration U extends to a map U : S → S , and then belongs to a class of π3(S ) = Z indexed by an integer B ∈ Z, called the baryon number. B is also the degree of the map U which can be explicitly calculated as 3 1 3 B ≡ B(x)d x =− εij k Tr(RiRj Rk)d x, (4) 24π 2 where B(x) is the baryon density. Skyrmions are the static field configurations of minimal energy for each value of B. In the figures we plot level-sets of baryon density B(x). It is well known that the B = 1Skyrmion can be found using the so-called Hedgehog ansatz [1] UH (x) = cos f(r)I2 + i sin f(r)xˆ · τ , (5) where r =|x| and xˆ is the radial unit vector. f(r) is a real radial profile function satisfying f(0) = π, f(∞) = 0. This produces a solution with rotationally symmetric energy and baryon density. Fig. 1 shows a B = 1 Hedgehog Skyrmion coloured as in [7]. The centres of the white 2 + 2 = and black regions are where π1 π2 0 and π3 > 0, π3 <0 respectively. The centres of the red, green and blue regions are where π = 0 and tan−1 π2 = 0, 2π , 4π respectively. This is 3 π1 3 3 the colouring scheme used throughout this paper. An important feature of the Hedgehog ansatz is that an isorotation, U(x) → AU(x)A†, which acts as a rotation among the pion fields, is equivalent to a rotation in space, U(x) → U(D(A)x), = 1 † where D(A)ij 2 Tr(τiAτj A ) is the standard SO(3) rotation matrix corresponding to the SU(2) matrix A. This can be understood in terms of the coloured Skyrmions, where a spatial rotation can be ‘undone’ by a reordering of colours, i.e. an isorotation of the pion fields. A fur- ther use of the colouring is that it shows when two Hedgehog Skyrmions are in the attractive channel [13]. This is when the colours on the nearest, facing sides of two Skyrmions match. The B = 2Skyrmion plays an important role in the dynamics of two B = 1 Skyrmions. This Skyrmion is toroidal, and is shown in Fig. 2. 516 D. Foster, N.S. Manton / Nuclear Physics B 899 (2015) 513–526 Fig. 2. B = 2 Skyrmion. To relate the Skyrme model to nuclear physics, Skyrmions should be quantised. As the Skyrmions are energy minima they can be treated as rigid bodies rotating in space and isospace. For general Skyrmions, semi-classical quantisation is the quantisation of the time-dependent space and isospace rotations A2 and A1 in the rigid-body ansatz † U(t,x) = A1(t)U0(D(A2(t))x)A1(t) , (6) where U0(x) is a static solution.
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